Waves are:

    • Local
    • Spread out
    • Add via interference rules (constructive and destructive)
    • Are derivable assuming Newton’s laws and an elastic medium

 

So the bottom line is that although waves have interesting behavior there properties are not surprising.

 

 

We now make a giant leap and require that waves move from the phenomenological or perhaps derivable to the fundamental.

 

We require the basic elements in our theory, particle and interactions, to have characteristics similar to particles and characteristics similar to waves. These properties are simply “the way it is”.  We will take the perspective that it is not justifiable in terms of some overriding principle merely valid based on observation.  Taking this point of view is verified in practice to an almost undeniable degree.  Quantum mechanics, the theory that merges these features successfully has very few doubters and stands an very firm ground.

 

Particle nature: BULLETS

-        observations reveal that the particle of the theory always show up in the measurement as distinct chunks. You will never measure 1/2 of an electron or a partial photon.  You will always get either one or none!  To understand this we will need to be clear about the measurement process.  It will be important to make predictions based on what you measure.

Wave nature: Interference

-        The classic way to begin to understand the wave nature of  QM is the interference particles and waves impinging on two slits.  There will be an interference pattern in both cases.  Waves from the classical view point contribute an amplitude

 

There are two types of idealized problems that are usually introduced to provide examples of quantum behavior. 

  • Two slit experiment:  Feynman uses this example to introduce QM. There are several great websites that discuss the ideas.
  • Filters: Again Feynman devotes considerable time to the topic and several interesting web sites can be found under Stern Gerlach Spin filters e. g.

 

 

Double slit accumulation

http://www.upscale.utoronto.ca/PVB/Harrison/DoubleSlit/Flash/Histogram.html

 

http://www.upscale.utoronto.ca/GeneralInterest/QM.html

 

filters;

http://faraday.physics.utoronto.ca/PVB/Harrison/SternGerlach/Flash/SGInteractive.html

 

 

 

 

 

Our goal is to obtain an overview of the essential ingredients in QM. Here are some salient aspects:

 

  • Quanta: Observations or measurements will record many discrete quantities. For example, a detector will detect an electron or detect no electron but it will not detect 0.5 electrons.  The quantum nature of particles allows for a probability of detection that can span continuously the values from 0-> 1 but independent of the probability an actual measurement finds that an entire electron either present or not present.
  • Amplitudes: To account for the wave nature of QM amplitudes are used to describe “how much”. For example, for a two slit experiment, one needs to know the amplitude for the particle to pass through slit 1 and arrive at location y on the screen.  Amplitudes are used to describe:
    • the contribution of various sequences of events or paths to a final result,
    • the content of a state.

 

 

To determine the probability of an event one squares the amplitude.

 

  • Vector spaces: In order to incorporate the nature of QM correctly the states are viewed as constituting a vector space.  One might notice that classically light exhibits wave behavior.  A very interesting example of how light behaves is the impact of polarizing filters. An interesting result is the transmission of light through two perpendicular filters (zero transmission) when a third filter is inserted in between the two perpendicular ones.  In this case the transmission need not be zero. Although the result might be surprising it is understandable because of the vector nature of the electric field.  Quantum states are vectors in Hilbert space.  Some states are fundamentally different from other states (orthogonal) a particle at location x1 is not at all the same as a particle at location x2. However a particle at location x1 does and a particle of definite momentum do share or overlap properties. Both of these states may result in a particle being detected at x1. The behavior of QS is encapsulated in the mathematics of vector spaces.
    • Vector addition: The sum of two QS with weight factors (amplitudes) is a valid QS. You can any state A to any state B and the result is a valid QS.

 

 

    • Basis: There exists sets of QS that are linearly independent and from which any conceivable QS can be built using the above addition rule.
    • Inner product: An inner product can be introduced by adding a dual space and a product rule.

 

Familiar vector space of 3-d does not typically use the notion of a dual space when introducing the inner or dot product. However 4-d spacetime can be conveniently cast into the above form and the minus sign for the time component, when calculating length, can be introduced by allowing the space and its dual to have opposite signs for the t-component.

    • Orthonormal: using the inner product one can impose the condition of normality and orthogonality for a basis.

 

These are somewhat formal mathematical rules but they are essential.  If one can learn to formulate QM as vector space with amplitudes and bases, then the behavior of QS can be extracted and a certain intuition can be built.

 

Consider the earth orbiting around the sun. We know the interaction and can solve for the orbit.

 

From the force you calculate the earth’s orbit and with initial condition you know where it is at all times.

 

 

Given the Hamitonian H one can calculate the quantum wave function that is equivalent to the full QS.

 

Know how the atom will behave in terms of the wave function.

 

Suppose the problem is too difficult to solve because the interaction is more complicated.

 

For the two slit experiment you propagate a particle to slit 1 (the interaction) and then to the screen.

For this I can write down an amplitude to reach the screen by nature of the interaction.

 

 

One difference in QM is that I need to find all possible ways to reach the ending state

 

 

I can imagine a more complex system with a series of interactions.

My goal would be to calculate the amplitude to travel along any given path and then sum over all possible paths.  Feynman diagrams diagrammatically represent this deeply complicated mathematical process.  A first order diagram in some sense shows the interaction as a single step process, as if you can get a good result by correcting the trajectory one time classically. High order diagrams are similar to the multiple corrections required to find the asteroids final trajectory.  In addition to the expansion of the interaction Feynman diagrams represent the multitude of ways that quantum systems can travel. The amplitudes for each trajectory will be included and interference may result.

 

There are several aspects of QM we will need to understand as general concepts.

 

Quantum states form a vector space (Hilbert space).

  • A QS can be written as a linear combination of other QS.
    • Tuning forks can be represented as a string of local pressures (snaps)
    • Snaps can be decomposed using the Fourier transform (tuning forks)
  • Certain sets of states can form  a basis
  • As vectors one can compute an inner product.
    • Orthogonality: No overlap between QS. This implies that the states are chosen to be completely distinct in some way. There is a label that can be used to completely distinguish state 1 and state 2.
    • Overlap: State can share some characteristic. Two states may both have some likelihood of being at some location.

 

Operators may represent observables so their behavior and interpretation tells us how to think about particle properties.